How do I find the complete factored form of a polynomial with a degree of $3$ , having a leading coefficient of $2$ with some zeros $i$ and $1$ ?

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Solution

Determine the complete factored form of a polynomial:
It is known that $x = a$ is a zero if and only if $(x - a)$ is a factor.
If the standard form of the cubic polynomial has real coefficients, any zeroes occur in complex conjugate pairs, then $i$ is a zero and $- i$ also a zero.
As the leading coefficient is $2$ .
Therefore, the cubic polynomial in factored form can be written as:
$f (x) = 2 (x - 1) (x - i) (x + i)$
The polynomial can be formed by multiplying the terms:
$\begin{array}{rcl} f (x) & = & 2 (x - 1) (x - i) (x + i) \\ = & 2 (x - 1) (x^{2} - i^{2}) [∵ (a + b) (a - b) = (a^{2} - b^{2})] \\ = & 2 (x - 1) (x^{2} - (- 1)) \\ = & 2 (x - 1) (x^{2} + 1) \\ = & 2 x^{3} - 2 x^{2} + 2 x - 2 \end{array}$
Therefore, the obtained polynomial is $f (x) = 2 x^{3} - 2 x^{2} + 2 x - 2$ .

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